29 problems

Let $T$ be the rooted tree associated with iterated preimages of a rational function of degree two, and let an arboreal representation be the continuous action on this tree arising…

Let $\mathfrak{\mathcal{C}}_{0}$ be the arrangement of six lines whose normals are the columns of the matrix $M_{t}$ for $t=\pm2\pm\sqrt{5}\in\mathbb{P}^{1}\setminus U$, and let…

Let $g\geq 1$, let $\alpha=(\alpha_1,\dots,\alpha_n)$ be a partition of $2g-2$, let $\mathcal H_1(\alpha)$ be the space of unit-area surfaces in the corresponding stratum of abelia…

Zariski dense orbit conjecture. Either there exists $\alpha\in X(K)$ whose orbit under $\Phi$ is well-defined and Zariski dense in $X$, or there exists a non-constant rational func…

Let $\mathcal H(4)^{\mathrm{hyp}}$ denote the hyperelliptic connected component of the stratum of translation surfaces with one zero of order $4$. An algebraically primitive Teichm…

Gizatullin's conjecture. If $f^*$ has infinite order on $Pic({\cal X})$, then ${\cal X}$ should have an anti-canonical curve.

McMullen's hyperbolicity conjecture. The derivative of the mapping class $[h]$ at $[\rho]$ is hyperbolic: no eigenvalue has norm $1$.

Let $\mathbf{G}$ be a semisimple algebraic group defined over $\mathbb{Q}$, let $\Gamma$ be a subgroup of $\mathbf{G}_{\mathbb{R}}$ commensurable with $\mathbf{G}_{\mathbb{Z}}$, le…

Higher-dimensional affine-symbol conjecture. Every bounded weighted composition operator with nonvanishing weight has affine symbol. This would extend the affine-symbol result from…

Large Sets Conjecture. If $S$ is 2-large, then $S$ is large.

Let $\mathcal P$ be an Apollonian circle packing with symmetry group $\Gamma<\operatorname{PSL}_2(\mathbb C)$. For any circle $C\in \mathcal C_\Lambda^*$, consider its orbit…

A post-critically finite branched covering of the Riemann sphere may admit stable multicurves and a canonical decomposition in the sense of Pilgrim. A canonical Thurston obstructio…

Let $X$ be a rational surface and let $\varphi$ be an automorphism of $X$ of infinite order. An effective antipluricanonical divisor is an effective divisor representing a negative…

Let $\Sigma_{0,4}$ be the four-holed sphere and let $\operatorname{Rep}_{qf}(\Sigma_{0,4})$ denote its space of quasi-Fuchsian representations. Let $S_{0,0,0,0}$ be the character s…

Let $\mathcal{M}$ be an orbit closure of surfaces of genus $g$. Its rank is denoted by $\operatorname{rank}(\mathcal{M})$. Almost High Rank Conjecture. If … then $\mathcal{M}$ is a…

Conjecture on totally periodic curves. Every totally periodic curve on $\mathbb{P}^3$ is linear.

The Siegel–Veech and Lyapunov intersection formulas. The area Siegel–Veech constant and the sum of Lyapunov exponents are

Uniform boundedness conjecture. For each degree $d\geq 2$, there exists a constant $B=B(d)$ such that either

Let $C$ be a compact Riemann surface, and let the $n$-th Hitchin component be the connected component of … containing symmetric powers of Fuchsian representations. For a representa…

Let $G$ be a semisimple Lie group with a $\Theta$-positive structure, let $G/P_\Theta$ be its associated flag variety, and let $G_\Theta^{>0}$ denote the $\Theta$-positive semigrou…

Let $K$ be a number field, let $\mathcal{S}$ be a finite set of places, and write $K_{\mathcal{S}}=\prod_{v\in\mathcal{S}}K_v$ and $\mathcal{O}$ for the ring of $\mathcal{S}$-integ…

Let $Q(v_1,v_2,v_3):=2v_1v_3-v_2^2$ and … Let $\mathcal{C}\subset\operatorname{Gr}_2(\mathbb{R}^3)$ be the circle of isotropic subspaces, and let…

Let $L$ and $S$ be as in Corollary, and let … Let $\mathfrak{D}_S(L)$ denote the corresponding set of conditioned directional lattices, and let…

Algebraicity conjecture. The closure of every complex geodesic in moduli space is an algebraic subvariety.

Let $\mathcal{M}$ be an affine manifold in a stratum of translation surfaces. Its rank is the dimension of its tangent space projected to absolute cohomology divided by $2$, and…